Problem: $f(x) = \dfrac{ \sqrt{ x - 9 } }{ x^2 + 13 x + 40 }$ What is the domain of the real-valued function $f(x)$ ?
Explanation: $f(x) = \dfrac{ \sqrt{ x - 9 } }{ x^2 + 13 x + 40 } = \dfrac{ \sqrt{ x - 9 } }{ ( x + 8 )( x + 5 ) }$ First, we need to consider that $f(x)$ is undefined anywhere where the radical is undefined, so the radicand (the expression under the radical) cannot be less than zero. So $x - 9 \geq 0$ , which means $x \geq 9$ Next, we also need to consider that $f(x)$ is undefined anywhere where the denominator is zero. So $x \neq -8$ and $x \neq -5$ However, these last two restrictions are irrelevant since $9 > -8$ and $9 > -5$ and so $x \geq 9$ will ensure that $x \neq -8$ and $x \neq -5$ Combining these restrictions, then, leaves us with simply $x \geq 9$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid x \geq9\, \}$.